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With the natural definitions of addition and scalar multiplication x s is a linear space.
Quotient space. One can readily verify that with this definition congruence modulowis an equivalence relation onv. Definition let fbe a field va vector space over fandw va subspace ofv. In topology and related areas of mathematics a quotient space is the quotient set of a topological space under an equivalence relation which is equipped with the quotient topology that is the finest topology the topology that has the largest set of open sets that makes continuous the canonical. The quotient space is an abstract vector space not necessarily isomorphic to a subspace of.
Quotient spaces are also called factor spaces. However if has an inner product then is isomorphic to in the example above. In topology and related areas of mathematics the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology that is with the finest topology that makes continuous the canonical projection map the function that maps points to their equivalence classes. The quotient space x s has as its elements all distinct cosets of x modulo s.
In linear algebra the quotient of a vector space v by a subspace n is a vector space obtained by collapsing n to zero. A subset of is called open iff is open in. If denotes the map that sends each point to its equivalence class in the topology on can be specified by prescribing that a subset of is. The space obtained is called a quotient space and is denoted v n read v mod n or v by n.
This can be stated in terms of maps as follows. Forv1 v2 v we say thatv1 v2modwif and only ifv1 v2 w.
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